On the cross-correlation properties of large-size families of Costas arrays

TL;DR

This paper investigates the cross-correlation properties of large families of Costas arrays, providing bounds for their maximal cross-correlation values in various shift scenarios, which is important for multi-user communication systems.

Contribution

It introduces new bounds on the maximal cross-correlation of large Costas array families, including Welch Costas arrays and power permutations, for different shift parameters.

Findings

Bound on cross-correlation for Welch Costas arrays with arbitrary shifts.

Bound on cross-correlation of power permutations for specific shifts.

First nontrivial bounds for combined families including exponential Welch arrays and power permutations.

Abstract

Costas arrays have been an interesting combinatorial object for decades because of their optimal aperiodic auto-correlation properties. Meanwhile, it is interesting to find families of Costas arrays or extended arrays with small maximal cross-correlation values, since for applications in multi-user systems, the cross-interferences between different signals should also be small. The objective of this paper is to study several large-size families of Costas arrays or extended arrays, and their values of maximal crosscorrelation are partially bounded for some cases of horizontal shifts $u$ and vertical shifts $v$. Given a prime $p \geq 5$, a large-size family of Costas arrays over $\{1, \ldots, p-1\}$ is investigated, including both the exponential and logarithmic Welch Costas arrays. An upper bound on the maximal cross-correlation of this family for arbitrary $u$ and $v$ is given. We also show that the maximal cross-correlation of the family of power permutations over $\{1, \ldots, p-1\}$ for $u=0$ and $v \neq 0$ is bounded by $\frac{1}{2}+\sqrt{p-1}$. Furthermore, we give the first nontrivial upper bound on the maximal cross-correlation of the larger family including both exponential Welch Costas arrays and power permutations over $\{1, \ldots, p-1\}$ for arbitrary $u$ and $v=0$ that it equals $(p-1) / t$ where $t$ is the smallest prime divisor of $(p-1) / 2$ if p is not a safe prime and is at most $(p-1)^{\frac{1}{2}}+(p-1)^{\frac{1}{4}}+\frac{1}{2}$ otherwise.